# Topologies of continuity for Carath\'{e}odory delay differential equations with applications in non-autonomous dynamics

## TL;DR

This paper investigates various topologies of integral type for Carathéodory delay differential equations, establishing continuous dependence of solutions and semiflows on initial data, with applications to non-autonomous dynamics.

## Contribution

It introduces new strong and weak topologies of integral type and analyzes their impact on the continuous dependence of solutions in delay differential equations.

## Key findings

- Established continuous dependence of solutions on initial data.
- Provided conditions for equivalence of different topologies.
- Extended results to Sobolev phase spaces.

## Abstract

We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carath\'eodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carath\'eodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carath\'eodory delay differential equations when the considered phase space is a Sobolev space.